# Data sheet ----> state space representation

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#### EOJ

##### Member
If I have the following circuit how would I go about representing it in state space form, specifically the non-linear devices diodes, BJTs, and LEDs? Would I be correct in focusing on the non-linear devices linear acting region? Or do I take their V I relationship and linearize it?Thanks ahead of time?

#### spec

##### Well-Known Member
If I have the following circuit how would I go about representing it in state space form, specifically the non-linear devices diodes, BJTs, and LEDs? Would I be correct in focusing on the non-linear devices linear acting region? Or do I take their V I relationship and linearize it?Thanks ahead of time?
Hi EOJ,

I'm intrigued by your circuit and your question. When you say 'state space form', do you mean 'transfer function'?

In electronics (and engineering in general) you have two options:

(1) Produce a formula taking into account the component characteristics and thus define the transfer function and perhaps produce a graph. Simulators are widely used for this; LTSPICE is very popular, especially on ETO. LTSPICE is also free.

(2) Design the circuit so that the variations are taken out and the transfer function is simplified. A good example of this is the use of negative feedback to produce an amplifier with low distortion and defined gain. For example, a typical opamp may have an open loop gain varying from 100K to 1M, depending on individual sample, and also have relatively high distortion. But, by the use of negative feedback, an amplifier can be made which has an accurately defined gain, of say of 10, and vanishingly low distortion, regardless of the gain of any individual opamp sample.

In your circuit the current through the LEDs is not accurately defined and the exponential characteristic of the transistor is not very well linerized. Of course, you may be just switching the transistor on and off and the non linear transfer function may not be relevant. Perhaps you could describe what you would like the circuit to do?

spec

#### MrAl

##### Well-Known Member
If I have the following circuit how would I go about representing it in state space form, specifically the non-linear devices diodes, BJTs, and LEDs? Would I be correct in focusing on the non-linear devices linear acting region? Or do I take their V I relationship and linearize it?Thanks ahead of time?
Hi there,

State Space does not really beg the question, "Do i linearize or do i not linearize", it's just another mathematical representation of a system, which can be linear or not linear in and of itself, and still have a state space representation. It's more about just getting a system into the right form, the way it exists as is. Yes, you may choose to linearize a non linear system, but that's a separate question apart from the representation itself. Of course a linear system will map more simply and will then lend itself to conventional techniques and if the system is non linear then you'll probably have to resort to numerical techniques.

For example take a diode, in linear form we may just write:
i=v/K1+K2

which is a linearized version, or we may write:
i=K1*(e^(v/K2)-1)

One is 'considered' linear and one is non linear. They are just different ways to represent the diode in mathematical form.

What is not clear in the drawing however, is where the light for the photo transistors is coming from. Is it coming from the ambient or from the LED's, or both? That would be something that would change the representation quite a bit, in fact completely, so that would have to be found out or else you'd have to consider all three ways unless there are other possibilities too and then you'd have to do those cases also.

What would probably help here is to do a couple much simpler systems maybe with one or two inputs and outputs first, then go from there. The linear cases are easier so that would be a good place to start

#### EOJ

##### Member
Thank you both for the time you spent answering my question. I can't tell you how much I appreciate your time.

Spec,

I'm intrigued by your circuit and your question. When you say 'state space form', do you mean 'transfer function'?
I'm new to the space space term, but essentially it means putting the system into a a form using matrices of the form dx/dt = Ax + Bu and y = Cx +Du. Kinda like a transfer function but the state space model allows for a better understanding about the system than the transfer function offers.

What is not clear in the drawing however, is where the light for the photo transistors is coming from. Is it coming from the ambient or from the LED's, or both?
The goal for me is to slightly modify the circuit above (put all IR LEDs in series and use a BD 140 instead of PN100 as Q9) and use it as an IR radar that will track either a free falling object. I intend on modulating the IR LEDs (pulsing using a 555 timer in its astable multivibrator config).Modulating and using some sort of lens will achieve a longer transmitting range, hopefully. Based on how much IR signal I receive back will be fed into differential op-amps that will drive two DC motors that will "track" the object. I know the object we want to track has to not absorb the IR.

Before I get into building I want to figure out how to represent this tracking system mathematically so I can model and simulate in MALAB/SIMULINK but I'm having trouble figuring out how to take a nonlinear device or nonlinear circuit and put it into a transfer function and state space representation.

#### MrAl

##### Well-Known Member
Hello again,

So you are saying that the light output from the IR LEDs will shine outward and the photo transistors will get that light unless the object blocks the light and then then get less light input?
And you intend to use AC modulation to avoid interference from other light sources?

You can probably start by just assuming that all the gains are constant, such as the photo transistor gains with light as input and current as output, and the gains of the LEDs are constant with current as input and light intensity as output.

I dont see any derivatives here though, what do you expect to have any derivatives for?
In the simpler circuit there is no energy storage. Adding energy storage seems like a waste unless you expect very large delays from the photo transistors, as the LEDs will have very little delay.

Heck, this writes out almost like a simple DC gain type circuit What are you doing with the motors though? If you need to model the exact response that can get more involved as then you might need the derivatives.

If you want to use state space you should really start with something simpler. Wikipedia gives some decent examples.

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